## Monday, September 14, 2009

### Dynamic Stability Of Bicycle Design : Part 2

Continued from Part 1.

Here, we'll study some fundamental concepts associated with bicycle motion, before we step into play mode. This is necessary for understanding what is to follow in later parts.

A bicycle is a single track vehicle and its dynamics can be studied with or without a rider. For purposes of our discussion here, we consider the human to be passive, rigidly attached to the rear frame of the bicycle, providing no control feedback whatsoever.

Fig 1 : A diagram of the model. Courtesy : Koojiman et.al

Engineers and scientists love to make models to study behaviors of systems. And it turns out that we can go ahead and make a mathematical model (see above) of the bicycle with a rear frame, a front frame with handlebars, and finally two wheels. Certain assumptions are made in the process, such as frictionless revolute joints, non-slipping rolling contacts for tires and knife-edge wheels.

Researches such as Francis Whipple and others have done this for almost a century, hence we stand on the shoulders of giants. The state of the art in bicycle dynamics also adopts the model and the linearized analysis behind it happens to be experimentally verified to justify the assumptions made in creating the model.

After introducing non-holonomic rolling and kinematic constraints, the typical 24 dimensional bicycle can be reduced to 3 to represent its configuration space. These are :

1) The roll rate of the rear frame, ф* (phi dot)
2) The steering rate, δ* (delta dot)
3) The angular rate of the rear wheel relative to the rear frame, θ* (theta dot)

Just imagine how complex analyzing bicycle dynamics in 3 dimensions is , leave alone analyzing it as a 24 dimensional system.

As it turns out, the state of the art model of the bicycle has 25 different design parameters, like the real world bicycle. These are shown in the graphic below. Just count the tick marks as you go along.

Fig 2 : Parameters affecting bicycle motion

Here, we get an idea of the different things that affect bicycle design. Its not just trail, or this angle, or that length, or this mass, but a picture bigger than that. Moreover, we can infer that there is an inter dependability among parameters. For example, changing the moment of inertia of your bicycle wheel is likely to change its mass as well.

In terms of their physical significance, single track vehicles possess 3 main modes of motion. There are precise scientific terms for these modes and its important that one doesn't muddle up their definitions and meanings.

1. CAPSIZE

Capsize mode is a non-oscillatory behavior involving both roll and steer and its prominence depends, among others, on the bicycle's speed and deceleration of the bike. The forward speed at which capsize motion neither grows nor decays is called capsize speed.

Basically, the mode tells you when the bicycle will lean over and fall and how easily it will do this. A bicycle without a rider at very low speed is unstable in roll and will simply fall to the ground laterally after moving into a tightening progressive spiral, sort of like a broomstick upon the action of gravity. A rider with some basic skills can easily stabilize this mode.

If a bicycle is rapidly decelerated by locking up the front wheel, it could capsize. In cornering at higher speeds, the ease with which capsize occurs (if you would consider it a rider controlled capsize) determines the cornering maneuverability of the bike. If capsize mode has a lesser time constant (less falling time), you can lean into turns and execute curves a little more correctly. So the lesser the falling time, the more unstable is this capsize mode.

Thus, we see how taking some stability away from the bike affords maneuverability. An overly stable bike is sluggish to control. Its not very responsive.

2. WEAVE

Weave is a complex, oscillatory behavior, 2-3 Hz in frequency, in which the bicycle oscillates or steers sinuously around the axis of the ground in the headed direction. The forward speed at which this oscillatory motion neither grows nor decays is called weave speed. This, although separate from the idea of the high speed shimmy or wobble cyclists always talk about, has a component of wobble in it that it is difficult to say which is which. These two modes are associated with each other in reality, although we like to think of them as separate.

Here's a video demonstration of weave I obtained from the net :

3. WOBBLE

Wobble is an unstable, oscillatory, steering motion. It can also be called steering oscillation. In popular literature, it is called Shimmy, a word that originates from an American dance style in the 1920's.

Here's a video demonstration of wobble. Use it to differentiate from weave.

From observations and anecdotal evidence, it is widely agreed that this mode occurs at some low speeds (see weave above) and also comes into play at some critically high speeds when frequencies are rapid from 5-9 Hz in range. To put things into perspective, a baby being rocked to sleep is at about 1 Hz. 9 Hz or more is rapid and dangerous and can quickly lead to a loss of control unless the rider consciously reverses the negative damping through body movements or braking.

The problem with rider provided damping is that sometimes, high speed wobble can be so quickly induced by some external disturbance that it takes the unassuming rider by surprise. This disturbance could arise from an irregularity on the road, or a bad mass of air, such as the wake turbulence from a box truck passing a cyclist on a descent. This initial condition could become large quickly before the rider can even react appropriately. The self-excited growth of this oscillation could lead to catastrophe.

From decades of detailed studies in motorcycle and airplane wobbles, researchers have agreed that a study of high speed shimmy is one involving the study of the elasticity of the steering head and frame and the complex interactions that come into play at the tire-ground interface.

When someone tells you that your loose bearings are what's causing the shimmy and you are completely sure that there are no loose bearings after periodic inspections, its time to expand your curiosity to the flexibility of the front end of the bike as well as the type and condition of the tyre you're using.

Almost all vehicles have the shimmy problem. It seems to be one of the engineering challenges in transport. A well designed bicycle is one in which the natural frequency of wobble is well above the speeds at which people normally travel on a bicycle. But meeting this is a challenge as bicyclists often like to mix and match different products and components while building bikes. Perhaps it would be wiser on the cyclist's part to keep the idea of a restricted speed range in mind while enjoying high speeds.

At this point, I'd like to shift your attention to the topic of this series. It is bicycle stability. In the next post, we will look at the bicycle parameters of our state of the art model and see how changing the values of the bicycle's parameters as shown in Fig 2 influence dynamic stability.

Keep the cup of your favorite beverage ready. And the rubber side down.